Tartarus, I understand, is part of the Greek mythological cosmology where souls are tormented in a dark pit or abyss far below even Hades.

Is the use of the term here by Peter indicative of an adoption of Hellenistic ideas of the underworld? (I.e. did Peter follow somewhat a Greek conception of the afterlife?) Or is it merely an appropriation of Hellenistic language to make a point to a particular audience?

It is used in Hesiod’s Theogony as the prison of the ancient Greek deities (the Titans); this is a classical Greek story with conceptual parallels to Gen 6:1–4. Tartarus is thus a place of extreme torment, in contrast to Elysium, the place of the blessed.2 In 2 Peter the name is used of the infernal region to which the rebellious angels were consigned, and hence here signifies a place of punishment of the wicked.3 According to another commentary,

One of the most prominent themes of ancient Jewish tradition, though
usually suppressed by the later rabbis, was the idea that the “sons of
God” in Genesis 6:1–3 were angels who lusted after women and so fell.
The term for “cast into hell” here is from the Greek name Tartarus, a
place not only of holding for the wicked dead (and especially the
Titans, the pre-Olympian supernatural beings), but of the severest
conceivable tortures; it occurs elsewhere in Jewish literature as the
place where the fallen angels were imprisoned. Jewish writers also
generally affirmed a current hell as a holding place for the wicked
until the final judgment.4

Peter may have intentionally wrote in such a way as to allude both to pagan mythology as well as the biblical narrative.

2 Peter is a piece of apology and polemic, responding to a crisis in
the church over God’s theodicy and the eschatological doctrine of the
Parousia as the end of the world and its judgment. First, 2 Peter
claims that heretics are already in the church: ‘false prophets’ who
speak peace when doom is coming (2:1-3) and ‘scoffers’ who mock ‘the
promise of his coming’ (3:3-4). They argue from the delay of the day
of judgment that God will not judge; from the eternity of the world
they argue against its predicted end.

Second, in response to this heresy, 2 Peter defends God’s coming
judgment, appealing to images intelligible to pagans and Christians
alike. The author affirms that ‘God did not spare’ the evil angels
(2:4), Noah’s world (2:5), or Sodom and Gomorrah (2:6-8). As God once
destroyed the world by water, so he can end it by fire (3:5-7). The
biblical allusions are clear, but these examples could also be
understood by pagans as references to their traditional myths of the
Titans cast into Tartarus, the flood of Deucalion and Phyrra, and the
fiery destruction of Phaethon. From these examples, 2 Peter concludes
with the principle that God both rewards and punishes: ‘God knows how
to rescue the godly…and to keep the unrighteous under punishment until
the day of judgment’ (2:9-10). Third, he defends God’s alleged
‘slowness’ in judging. God’s time is mysterious, as the psalmist noted
(3:8). God’s ‘slowness’ is really God’s long-suffering, giving sinners
time to repent (3:9). Even Paul, notoriously difficult to understand,
agrees with 2 Peter on God’s slowness to judge as God’s gift of
long-suffering (3:15-16; see Rom. 2:4-6).5

Based not only on the use here of ταρταρόω6, but also on the near universal acceptance of New Testament writers of using ᾅδης (Hades) as a translation for שְׁאוֹל (Sheol, although ᾅδης is not used by Peter), it seems very plausible that the early Christians used Hellenistic terms and concepts concerning the after-life but assigned to them slightly altered meanings from Jewish thought.

1William Arndt, Frederick W. Danker and Walter Bauer, A Greek-English Lexicon of the New Testament and Other Early Christian Literature, 3rd ed. (Chicago: University of Chicago Press, 2000), 991.